Spin-orbit coupling (SOC) is a relativistic effect whose origin lies in the Dirac’s equation – a relativistic analogue of Schrödinger’s equation. SOC corrects the electronic states of a quantum mechanical system up to ~1 eV in case of semiconductors and ~ 2 – 3.6 eV in case of actinides and heavy elements by considering not only the coordinates but also the spin of the electrons in the system. Most of the applications of the present day technology are based on manipulating the electronic structure of a system with very high accuracy and precision. This demands availability of correct electronic structure of a material or molecule within a feasible computational time. Some direct consequences of SOC in materials can be noticed in analyzing the charge-transport properties of a semiconductor, evaluating the candidature of transition metal dichalcogenides (TMDCs) for spintronic, twistronic and valleytronic applications, and in the origin of topological properties of a material. Not only in materials but also in molecules the SOC effects can be observed. Fine-structure of atomic spectra was explained on the account of SOC. Several additional peaks and wavelength shift in UV-vis spectroscopy of Gold Superatoms can only be explained by correctly considering the energy level splittings caused by SOC. SOC allows intersystem and reverse intersystem crossing by mixing the spin states, ultimately opening various chemical reaction pathways which were spin forbidden before. Current advancements in computational power enrich us to work shoulder to shoulder with experiments where one can simulate the synthesized structures containing thousands of atoms using semi-empirical methods as in DFTB, GFN-XTB. These methods so far considered SOC effects but only as case studies in testing the implementation of SOC Hamiltonian rather than a systemic extension of SOC parameters to most part of the periodic table and studying SOC effects for different categories of materials and molecules. This motivated us to implement the SOC either in the form of highly accurate parameters throughout the periodic table or as addition in hamiltonian in such methods. Twisted van der Waals 2D materials as in twisted TMDC bilayers shows exciting electronic and optoelectronic properties and depending on the twist angle and chemical composition they can have thousands of atoms in their superlattices. A correct electronic analysis of such structures with SOC corrected DFT is computationally very expensive but is feasible at semi-empirical level. Here, we have applied our implementation on TMDC homo and heterobilayer twisted superlattices and studied the effect of SOC on the excitonic properties of the system. Therefore, this work opens the way for realizing various exotic applications of present day materials as well as molecules.:Table of Contents Abstract 4 1 Introduction 8 1.1 Quantum Chemistry: 8 1.2 HF based Semi-Empirical Methods 9 1.3 DFT based Semi-Empirical Methods 11 1.3.1 Density Functional based Tight-Binding Method (DFTB) 11 1.3.2 Geometry, Frequency, Non-Covalent, extended Tight Binding (GFN-xTB) 12 1.4 Spin-Orbit Coupling (SOC) 14 1.4.1 SOC in Materials 18 1.4.2 SOC in Molecular Structures 22 1.5 Theoretical Models for Accounting SOC 24 1.6 Motivation, Objective and Outline of thesis 26 2 Methodology 29 2.1 Quantum Chemistry 30 2.1.1 Schrödinger equation 30 2.2 Density Functional Theory 33 2.2.1 Generalized Gradient Approximations 39 2.3 Spin-orbit Coupling (SOC) 41 2.3.1 Classical Picture of SOC in LS model 42 2.3.2 Quantum Picture of SOC in LS model: 43 2.3.3 Calculation of SOC Paramentes 45 2.4 Density Functional Based Semi-empirical Quantum Mechanical Methods 48 2.4.1 Self-Consistent Charge Density Functional Based Tight Binding Method (SCC-DFTB) 48 2.4.2 Extended Tight-Binding (GFN1-xTB) 51 2.4.3 Addition of Spin-Orbit Coupling Hamiltonian in DFTB and GFN-xTB 54 3 Benchmarking Spin-Orbit Coupling Parameters for DFTB 56 3.1 Introduction 58 3.2 Computational Details of the DFT benchmark calculations 60 3.3 Benchmarking Spin-Orbit Coupling Parameters 60 3.3.1 III-V Bulk Semiconductor 61 3.3.2 Transition Metal Dichalcogenide 2D Crystals 65 3.3.3 Topological Insulators 68 3.4 Conclusions 70 4 Spin-Orbit Coupling Corrections for the GFN-xTB method 71 4.1.1 Introduction 73 4.2 Computational Details of The Benchmark Calculations 75 4.3 Results & Discussion 76 4.3.1 Geometries 76 4.3.2 Effect of SOC on Charge Transport Properties of Chromophores in MOFs 77 4.3.3 Superatoms 82 4.3.4 Effect of SOC on Binding of O2 on Ferrous Deoxyheme 85 4.4 Conclusions 86 5 Spin Orbit Coupling Effects on The Excitonic Properties of Twisted Moiré Transition Metal Dichalcogenides 88 5.1 Introduction 90 5.2 Computational Details 92 5.3 Results & Discussions 93 5.4 Excitons in Twisted Moiré Homobilayers 93 5.5 Excitons in Twisted Moiré Heterobilayers 102 5.6 Conclusions 109 6 Summary 112 A. Acronym 116 B. Appendices 120 SOC Parameters 120 7 References 147 C. Acknowledgement 173

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京都大学 / 新制・課程博士 / 博士(エネルギー科学) / 甲第24116号 / エネ博第449号 / 新制||エネ||84(附属図書館) / 京都大学大学院エネルギー科学研究科エネルギー応用科学専攻 / (主査)教授 大垣 英明, 教授 松田 一成, 教授 宮内 雄平 / 学位規則第4条第1項該当 / Doctor of Energy Science / Kyoto University / DFAM

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The question posed in this work is how one would model and predict the rotational spectrum of open-shell open-shell van der Waals complexes. There are two secondary questions that arise: the nature of radical-radical interactions in such systems and the modelling of the large amplitude motion of the constituent molecules. Four different systems were studied in this work, each providing part of the answer to the main question. Starting with the large amplitude motion, there are two theoretical approaches that may be adopted: to either model the whole complex as a semi-rigid molecule, or to perform quantum dynamical calculations. We recorded and analysed the rotational spectrum (using Fourier transform microwave spectroscopy) of the molecule of tertiary butyl acetate (TBAc) which exhibits a high degree of internal rotation; and of the weakly-bound complex between a neon atom and a nitrogen dioxide molecule (Ne-NO2). We used the semi-rigid approach for TBAc and the quantum dynamical approach for Ne-NO2. We also explored the compatibility of these two approaches. Moreover, we were able to predict and analyse the fine and hyperfine structure of the Ne-NO2 spectrum using spherical tensor operator algebra and the results of our dynamics calculations. To explore the nature of the interactions in an radical-radical van der Waals complex we calculated the PESs of the possible states that the complex may be formed in, when an oxygen and a nitrogen monoxide molecule meet on a plane using a number of high level ab initio methods. Finally, our conclusions were tested and applied when we performed the angular quantum dynamics to predict the rotational spectrum of the complex between an oxygen and a nitrogen dioxide molecule, and account for the effect of nuclear spin statistics in that system.

Dans cette thèse, on présente un nouveau type de source atomique: un jet superfin d'atomes métastables produit à partir de l'échange de métastabilité au sein d'un jet supersonique d'atomes de gaz rares. On a utilisé les propriétés de cohérence de ce jet pour observer la diffraction de l'hélium, de l'argon et du néon métastables par un nano-réseau par transmission et par des micro-réseaux par réflexion. Ensuite, on a mis en évidence des transitions entre sous niveaux Zeeman du niveau 3P2 du néon métastable, transitions dues au terme quadrupolaire de l'interaction de van der Waals entre l'atome et la surface. Après avoir montré les résultats expérimentaux, nous avons calculé les probabilités de transitions dans le modèle de Landau–Zener puis discuté de l'intérêt des transitions vdW-Zeeman pour l'interférométrie atomique. Enfin, on a décrit le procédé de ralentissement Zeeman du jet supersonique d'argon métastable (3P2) et montré les premiers spectres de temps de vol d'atomes ralentis obtenus.

This thesis, submitted to the University of Manchester, covers a range of topics related to current research in two-dimensional materials under the title: 'Investigations into the interfacial interaction of graphene with hexagonal boron nitride.'In the last decade, two-dimensional materials have become a rich source of original research and potential applications. The main advantage lies in the ability to produce novel composite structures, so-called 'layered heterostructures', which are only a few atomic layers thick. One can utilise the unique properties of several species of crystal separately, or how they interact to realise a diverse range of uses. Two such crystals are graphene and hexagonal boron nitride. Hexagonal boron nitride has, so far, been used primarily as a substrate for graphene, allowing researchers to get the most out of graphene's impressive individual properties. However, in this thesis, the non-trivial van der Waals interaction between graphene and hexagonal boron nitride is examined. The interface potential reveals itself as a relatively large-scale, orientation-dependant superlattice, which is described in chapters 1 and 2.I In Chapter 4, the effect of this superlattice is examined by measurement of its effect upon the electrons in graphene, where its modulation leads to the creation of second and third generation Dirac points, revealing Hofstadter's Butterfly. As well as an excellent example of the physics possible with graphene, it also presents a new tool with which to create novel devices possessing tailored electronic properties. II In chapter 5, the consequential effect of the superlattice potential on the structure of graphene is studied. Results are discussed within the framework of the Frenkel-Kontorova model for a chain of atoms on a static background potential. Results are consistent with relaxation of the graphene structure leading to the formation of a commensurate ground state. This has exciting consequences for the production of heterostructures by demonstrating that alignment angle can have large effects upon the physical properties of the crystals. III In chapter 6, the van der Waals potential is shown to be responsible for the self-alignment of the two crystals. This effect is important for the fabrication of perfectly aligned devices and may lead to new applications based on nanoscale motion.

This thesis presents several experiments, generally aiming at visualising the ballistic and topological transport on the high-mobility graphene/boron nitride heterostructure using the scanning gate microscope. For the first experiment, we use the scanning gate microscopy to map out the trajectories of ballistic carriers in high-mobility graphene encapsulated by hexagonal boron nitride and in a weak perpendicular magnetic field. We employ a magnetic focusing transport configuration to image carriers that emerge ballistically from an injector, follow a cyclotron path due to the Lorentz force from an applied magnetic field, and land on an adjacent collector probe. The local potential generated by the scanning tip in the vicinity of the carriers deflects their trajectories, modifying the proportion of carriers focused into the collector. By measuring the voltage at the collector while scanning the tip, we are able to obtain images with arcs that are consistent with the expected cyclotron motion. We also demonstrate that the tip can be used to redirect misaligned carriers back to the collector. For the second experiment, we investigate the graphene van der Waals structures formed by aligning monolayer graphene with insulating layers of hexagonal boron nitride which exhibit a moiré superlattice that is expected to break sublattice symmetry. However, despite an energy gap of several tens of millielectronvolts opening in the Dirac spectrum, electrical resistivity remains lower than expected at low temperature and varies between devices. While subgap states are likely to play a role in this behaviour, their precise nature is still unclear in the community. We therefore perform a scanning gate microscopy study of graphene moiré superlattice devices with comparable activation energy but with different charge disorder levels. In the device with higher charge impurity ($\sim$ 10$^-$ cm$^{-2}$) and lower resistivity ($\sim$ 10 k$\Omega$) at the Dirac point we observe scanning gate response along the graphene edges. Combined with simulations, our measurements suggest that enhanced edge doping is responsible for this effect. In addition, a device with low charge impurity ($\sim$ 10$^{9}$ cm$^{-2}$) and higher resistivity ($\sim$ 100 k$\Omega$) shows subgap states in the bulk. Our measurements provide alternative model to the prevailing theory in the literature in which the topological bandstructures of the graphene moiré superlattices entail an edge currents shunting the insulating bulk. In the third experiment, we continue our study in the graphene moir$\acute e$ superlattices with the newly reported non-local Hall signals at the main Dirac point. It has been associated with the non-zero valley Berry curvature due to the gap opening and the nonlocal signal has been interpreted as the signature of the topological valley Hall effects. However, the nature of such signal is still disputed in the community, due to the vanishing density of states near the Dirac point and the possible topological edge transport in the system. Various artificial contribution without a topological origin of the measurement scheme has also been suggested. In connection to the second experiment, we use the scanning gate microscope to image the non-local Hall resistance as well as the local resistance in the current path. By analysing the features in the two sets of images, we find evidence for topological Hall current in the bulk despite a large artificial components which cannot be distinguished in global transport measurement. In the last experiment, we show the development of a radio-frequency scanning impedance microscopy compatible with the existing scanning gate microscopy and the dilution refrigerator. We detailed the design and the implementation of the radio-frequency reflectometry and the specialised tip holder for the integration of the tip and the transmission lines. We demonstrate the capability of imaging local impedance of the sample by detecting the mechanical oscillation of the tip, the device topography, and the Landau levels in the quantum Hall regime at liquid helium temperature and milli-Kelvin temperature.

Dans cette thèse, on a montré l'occurrence de transitions exothermiques et endothermiques entre les niveaux métastables 3P_2 et 3P_0 dans la collision d'atomes de gaz rare avec une surface solide. Le développement d'une source d'atomes métastables utilisant le processus d'échange de métastabilité au sein d'un jet supersonique a permis ensuite d'observer la diffraction par un réseau de nanofentes, puis, dans une expérience de collision sur un réseau micrométrique en Cuivre en présence d'un champ magnétique externe, d'observer des transitions Zeeman au sein du niveau 3P_2. Le calcul, à partir des données spectroscopiques, de l'interaction de van der Waals entre un atome d'argon métastable dans l'état 3P_2 et une surface conductrice plane a révélé outre une partie scalaire, une partie quadrupolaire modifiant l'énergie des sous niveaux au voisinage de la surface. Un modèle d'évolution soudaine prédit alors une probabilité de transition dont l'ordre de grandeur est en accord avec l'expérience.